The octonions are a non-associative extension of the quaternions. They were discovered by John T. Graves in 1843, and independently by Arthur Cayley, who published the first paper on them in 1845. They are sometimes referred to as Cayley numbers or the Cayley algebra.
The octonions form an 8-dimensional algebra over the real numbers, and can therefore be thought of as octets of real numbers. Every octonion is a real linear combination of the unit octonions 1, e1, e2, e3, e4, e5, e6 and e7, the multiplication table for which looks as follows.
| · | 1 | e1 | e2 | e3 | e4 | e5 | e6 | e7 |
| 1 | 1 | e1 | e2 | e3 | e4 | e5 | e6 | e7 |
| e1 | e1 | -1 | e4 | e7 | -e2 | e6 | -e5 | -e3 |
| e2 | e2 | -e4 | -1 | e5 | e1 | -e3 | e7 | -e6 |
| e3 | e3 | -e7 | -e5 | -1 | e6 | e2 | -e4 | e1 |
| e4 | e4 | e2 | -e1 | -e6 | -1 | e7 | e3 | -e5 |
| e5 | e5 | -e6 | e3 | -e2 | -e7 | -1 | e1 | e4 |
| e6 | e6 | e5 | -e7 | e4 | -e3 | -e1 | -1 | e2 |
| e7 | e7 | e3 | e6 | -e1 | e5 | -e4 | -e2 | -1 |
The octonions are the only alternative but not associative finite-dimensional division algebra over the reals. The finite-dimensional associative division algebras are the reals, the complex numbers, and the quaternions.
See also Hypercomplex numbers.
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